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Airy wrote a paper (maybe deceptively) titled On certain Conditions under which a Perpetual Motion is possible (Cambridge Philosophical Transactions, December 14, 1829). Alternative site (in more than one sense) with JPEG images: https://www.besslerwheel.com/airy/. I've OCRed the (very legible) images and I can post the text here if required.

As a jack of all trades and master of none (I have read and understood The Theoretical Minimum by Susskind and Hrabovsky), I found the paper rather dense. Can somebody summarize it in modern language?

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    $\begingroup$ Precis: Air passing through a flexible reed can cause sound. Nothing about perpetiual motion except the misleading title! $\endgroup$
    – mike stone
    Commented May 27, 2020 at 17:53
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    $\begingroup$ Not really! The understanding as to how wind instruments make sound has developed a great deal since Airy, and his theory, although a nice bit diff-eq theory is quite obsolete. $\endgroup$
    – mike stone
    Commented May 27, 2020 at 19:06
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    $\begingroup$ The author of the paper is accurate in deducing the effects of his postulated law of force in the particular phenomenon. However on page 3 they note that the postulates are not backed up by experiment. $\endgroup$
    – kbakshi314
    Commented May 27, 2020 at 19:15
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    $\begingroup$ Re I found the paper rather dense -- Try reading Newton's Philosophiæ Naturalis Principia Mathematica, which he wrote in Latin. Every English translation looks like Greek to me. Or try reading James Maxwell's A Treatise on Electricity and Magnetism. While written in English, it is still ... Greek. Almost all profession writing regarding physics before the late 19th century will inevitably be a bit inscrutable because those writers did use modern mathematical notation. And almost all profession writing regarding physics since the late 19th century will be more than a bit dense. $\endgroup$
    – David Hammen
    Commented May 28, 2020 at 7:03
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    $\begingroup$ It's probably worth knowing there's a sibling question over at skeptics.stackexchange.com/questions/47693/… $\endgroup$
    – Kyle Miller
    Commented May 28, 2020 at 23:39

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He considers the differential equation $$\phi''(t)=-e \phi(t)-g\phi(t-c),$$ which models a body acted upon by a force whose magnitude depends on the body's "position at some time preceding that action" (quoted from the second paragraph). He is inspired by an analysis of vocal chords by someone named Mr. Willis.

If $g=0$, this is a run-of-the-mill second order linear differential equation, and it has the solution $$\phi(t)=a\sin(t\sqrt{e}+b)$$ for $a$ and $b$ arbitrary constants.

But with $g>0$ it is difficult to solve, and I'm unaware of a theorem that proves a solution even exists. He refers to another paper that (presumably; I didn't check it myself) gives a formula for a small-$g$ approximation, and then he describes, using their notation, how to calculate the increase in amplitude from cycle to cycle. This corresponds to an increase in energy in the system.

Commentary. A way to think about this differential equation is that it is modeling a driven simple harmonic oscillator without damping and with driving force $-g\phi(t-c)$. If $g$ is small, then in the short term the system has the characteristics of an undriven simple harmonic oscillator, thus operates at its natural frequency. The driving force is a phase-shifted version of its position, which in particular is at the natural frequency, and driving a simple harmonic oscillator at its natural frequency causes resonance.

The ability of the system to drive itself is probably where the energy is coming from, since the system is having to do work to apply this force through time. What I mean is, it takes external energy to apply this force.

In short: it is a paper about an approximate solution to a second-order differential equation that was inspired by some contemporaneous research.

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  • $\begingroup$ Thanks. And, according to a private comment, "Nothing about perpetual motion except the misleading title!". Do you agree? $\endgroup$
    – Martín-Blas Pérez Pinilla
    Commented May 28, 2020 at 23:36
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    $\begingroup$ @Martín-BlasPérezPinilla I can't say without knowing more context about what they were thinking about back then. For all I know, Airy's paper is a polite way of saying Willis's proposal that vocal chord motion involves a force depending on their past position should be dismissed because such a law would lead to perpetual motion. $\endgroup$
    – Kyle Miller
    Commented May 28, 2020 at 23:45
  • $\begingroup$ "... a polite way of saying Willis's proposal..." +1000! (factorial, not exclamation). $\endgroup$
    – Martín-Blas Pérez Pinilla
    Commented May 29, 2020 at 5:22

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